On Hilbert space, \(\hat{G}\)  is a metric and coincides with the operator norm of the difference between orthogonal projections.
On Banach space, \(\hat{G}\)  is not a metric since it does not satisfies the triangle inequality in general [1]}.
However, the gap \(\hat{G}\)  and the distance \(d_{H}\)  are related by the following proposition (see [1]} or [3]}):
\(\hat{G}(E,F) \le d_{H}(E,F) \le 2\hat{G}(E,F).\) 
